大尺度约简原理及其在假设检验中的应用
Large scale reduction principle and application to hypothesis testing
摘要：
考虑一个非线性函数$G(X_t)$，其中$X_t$是具有长程相关性的平稳高斯序列.通常的归约原理指出，$G(X_t)$的部分和渐近地表现为$G$在Hermite多项式中的展开式中的首项的部分和。在长程相关参数的小波估计的上下文中，用小波尺度图代替$G(X_t)$的部分和，即小波系数的平方的部分和。在小波设置中是否存在一个约化原理，即$G(X_t)$的尺度图的渐近行为是否与$G$在Hermite多项式中展开的首项的渐近行为相同？总的来说，答案是否定的。本文提供了一个最小的增长条件的小波系数的尺度，以确保减少原则也适用于尺度图。结果被施加到测试的假设，长程相关参数采取一个特定的值。
英文摘要：
Consider a non--linear function $G(X_t)$ where $X_t$ is a stationary Gaussian sequence with long--range dependence. The usual reduction principle states that the partial sums of $G(X_t)$ behave asymptotically like the partial sums of the first term in the expansion of $G$ in Hermite polynomials. In the context of the wavelet estimation of the long--range dependence parameter, one replaces the partial sums of $G(X_t)$ by the wavelet scalogram, namely the partial sum of squares of the wavelet coefficients. Is there a reduction principle in the wavelet setting, namely is the asymptotic behavior of the scalogram for $G(X_t)$ the same as that for the first term in the expansion of $G$ in Hermite polynomial? The answer is negative in general. This paper provides a minimal growth condition on the scales of the wavelet coefficients which ensures that the reduction principle also holds for the scalogram. The results are applied to testing the hypothesis that the long-range dependence parameter takes a specific value.
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